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Status: MACHINE-VALIDATED FINITE INSTANCE

This page records a first-principles nonlinear-string model in which a localized relational pattern propagates, retains measurable structure, reaches a terminal region, and delivers positive work to a load.

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Research question

Can a finite dissipative substrate transport a recursively identifiable pattern while performing measurable positive work without immediately erasing that pattern?

Closed model

For sites $n=0,\ldots,N-1$, define

$$ m\ddot q_n+\gamma\dot q_n+\alpha q_n^3+\kappa(2q_n-q_{n-1}-q_{n+1})=F_n(t)+L_n(t). $$

The left endpoint receives a finite Gaussian-windowed sinusoidal drive. The terminal site is coupled to a passive spring-damper load:

$$ L_{N-1}(t)=-k_Lq_{N-1}-c_L\dot q_{N-1}. $$

The validated run used $N=64$, $m=1$, $\gamma=0.018$, $\kappa=1$, $\alpha=0.16$, $k_L=0.45$, and $c_L=0.08$.

First-principles observables

Local energy

$$ e_n(t)=\frac12m\dot q_n^2+\frac14\alpha q_n^4+\frac14\kappa(q_n-q_{n-1})^2+\frac14\kappa(q_{n+1}-q_n)^2. $$

At the terminal site, the stored load-spring term $\tfrac12k_Lq_{N-1}^2$ is included.

Energy centroid

$$ x_E(t)=\frac{\sum_n n e_n(t)}{\sum_ne_n(t)}. $$

Shift-invariant pattern persistence

Let $p_0$ be the normalized local-energy profile at peak injected energy and $p(t)$ the normalized profile at time $t$. Define

$$ \Pi(t)=\max_s\frac{\langle T_sp_0,p(t)\rangle}{\|T_sp_0\|\,\|p(t)\|}, $$

where $T_s$ is a zero-padded spatial translation.

First-passage transport velocity